Properties

Label 2-162-27.23-c2-0-5
Degree $2$
Conductor $162$
Sign $-0.979 - 0.202i$
Analytic cond. $4.41418$
Root an. cond. $2.10099$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.483 − 1.32i)2-s + (−1.53 − 1.28i)4-s + (−7.52 − 1.32i)5-s + (4.40 − 3.69i)7-s + (−2.44 + 1.41i)8-s + (−5.40 + 9.35i)10-s + (−8.02 + 1.41i)11-s + (−22.0 + 8.02i)13-s + (−2.78 − 7.64i)14-s + (0.694 + 3.93i)16-s + (−6.39 − 3.69i)17-s + (−7.80 − 13.5i)19-s + (9.82 + 11.7i)20-s + (−2.00 + 11.3i)22-s + (19.9 − 23.8i)23-s + ⋯
L(s)  = 1  + (0.241 − 0.664i)2-s + (−0.383 − 0.321i)4-s + (−1.50 − 0.265i)5-s + (0.629 − 0.528i)7-s + (−0.306 + 0.176i)8-s + (−0.540 + 0.935i)10-s + (−0.729 + 0.128i)11-s + (−1.69 + 0.617i)13-s + (−0.198 − 0.546i)14-s + (0.0434 + 0.246i)16-s + (−0.376 − 0.217i)17-s + (−0.410 − 0.711i)19-s + (0.491 + 0.585i)20-s + (−0.0909 + 0.515i)22-s + (0.868 − 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.979 - 0.202i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ -0.979 - 0.202i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0536046 + 0.524963i\)
\(L(\frac12)\) \(\approx\) \(0.0536046 + 0.524963i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.483 + 1.32i)T \)
3 \( 1 \)
good5 \( 1 + (7.52 + 1.32i)T + (23.4 + 8.55i)T^{2} \)
7 \( 1 + (-4.40 + 3.69i)T + (8.50 - 48.2i)T^{2} \)
11 \( 1 + (8.02 - 1.41i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (22.0 - 8.02i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (6.39 + 3.69i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (7.80 + 13.5i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-19.9 + 23.8i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-9.68 + 26.6i)T + (-644. - 540. i)T^{2} \)
31 \( 1 + (-12.2 - 10.2i)T + (166. + 946. i)T^{2} \)
37 \( 1 + (-5.99 + 10.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (2.82 + 7.76i)T + (-1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (7.82 + 44.3i)T + (-1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-43.4 - 51.7i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 + 16.4iT - 2.80e3T^{2} \)
59 \( 1 + (57.3 + 10.1i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-54.6 + 45.8i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (47.1 - 17.1i)T + (3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (35.4 + 20.4i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (49.7 + 86.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (72.4 + 26.3i)T + (4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (36.0 - 99.1i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (-0.302 + 0.174i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-11.1 - 62.9i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.05296861184962034385076349919, −11.24350121741064573077433517557, −10.39743500011622383488066813926, −9.026219749348223006281969217976, −7.898229947735884278673805227933, −7.04916678191920289544937659512, −4.80017248181638221933158008503, −4.37502226723569762904431854185, −2.61855057859497439494329598818, −0.29471203040108868466478866246, 2.96751981334847991691857823725, 4.47319276917107690516235856889, 5.44441967934657788830281037051, 7.15920892762303890996300847707, 7.81147500240728168574389120512, 8.645516575517505728939899040366, 10.20004423346863681728597562521, 11.40842957806401609831476428617, 12.17984732034835424617456306886, 13.05292266844272751874745848358

Graph of the $Z$-function along the critical line